S 8-Regularity Theorem for Nondegenerate CR Mappings

Monatsh. Math. 142, 315–326 (2004) DOI 10.1007/s00605-003-0053-2

A C 1 -Regularity Theorem for Nondegenerate CR Mappings
By

Bernhard Lamel Universit€at Wien, Austria Received November 26, 2002; in revised form June 24, 2003 Published online February 16, 2004 # Springer-Verlag 2004 0

Abstract. We prove the following regularity result: If M  CN , M 0  CN are smooth generic submanifolds and M is minimal, then every C k -CR-map from M into M 0 which is k-nondegenerate is smooth. As an application, every CR diffeomorphism of k-nondegenerate minimal submanifolds in CN of class C k is smooth. 2000 Mathematics Subject Classification: 32H02 Key words: Nondegenerate CR-mapping, regularity, finitely nondegenerate submanifolds

1. Introduction and Statement of Results We first briefly describe the setting for the results which we want to discuss. 0 0 Let M  CN , M 0  CN be generic, real submanifolds of CN and CN , respectively. We shall denote by d the real codimension of M and by d 0 the real codimension of M 0 , and write n ¼ N  d, n0 ¼ N 0  d 0 . Recall that M is generic if there is a smooth defining function
¼ ð
1 ; . . . ;
d Þ for M such that the vectors
1;Z ðpÞ; . . . ;
d;Z ðpÞ are linearly independent for p 2 M. Here for any smooth func@ @ tion  we let Z ¼ ð@Z ; . . . ; @Z Þ be its complex gradient. 1 N We also fix points p0 2 M and p00 2 M 0 (which we will assume to be equal to 0 for most of this paper). A C k -mapping H from M into M 0 is said to be CR if its c differential dH satisfies dHðTpc MÞ  THðpÞ M 0 for p 2 M, where Tpc M denotes the complex tangent space to M at p, that is, the largest subspace of the real tangent space Tp M invariant under the complex structure operator J in CN . Equivalently, if 0 H ¼ ðH1 ; . . . ; HN 0 Þ for any system of holomorphic coordinates in CN , each Hj is a CR-function on M. (For further reference on these definitions, the reader is referred to the book of Baouendi, Ebenfelt and Rothschild [1]). The following definition is from [9]. We shall give it in a slightly modified form. Definition 1. Let M, M 0 be as above. Let
0 ¼ ð
01 ; . . . ;
0d0 Þ be a defining function for M 0 near Hðp0 Þ, and choose a basis L1 ; . . . ; Ln of CR-vector fields tangent to M near p0. We shall write L ¼ L1 1    Ln n for any multiindex . Let H : M ! M 0
The author would like to thank the Erwin Schr€ odinger Institut in Vienna for its hospitality.

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be a CR-map of class0 C m . For 0 4 k 4 m, define the increasing sequence of subspaces Ek ðp0 Þ  CN by Ek ðp0 Þ ¼ spanC fL
0l;Z 0 ðHðZÞ; HðZÞÞjZ¼p0 : 0 4 jj 4 k; 1 4 l 4 d 0 g:

ð1Þ

We say that 0 H is k0 -nondegenerate at p0 (with 0 4 k0 4 m) if Ek0 1 ðp0 Þ 6¼ Ek0 ðp0 Þ ¼ CN . The invariance of this definition under the choices of the defining function, the basis of CR vector fields and the choices of holomorphic coordinates in CN and N0 C is easy to show; the reader can find proofs for this in [9] or [8]. Recall that if
 Rd is an open convex cone, p0 2 M, and U  CN is an open neighbourhood of p0 , then a wedge W with edge M centered at p0 is defined to be a set of the form W ¼ fZ 2 U :
ðZ; Z Þ 2
g, where
is a local defining function for M. We can now state our main theorem. 0

Theorem 2. Let M  CN , M 0  CN be smooth generic submanifolds of CN N0 and C , respectively, p0 2 M and p00 2 M 0 , H : M ! M 0 a Ck0 -CR-map which is k0 nondegenerate at p0 and extends continuously to a holomorphic map in a wedge W with edge M. Then H is smooth in some neighbourhood of p0 . This theorem is the smooth version of the main result in [9]. Let us recall that M is said to be minimal at p0 if there does not exist any CR-submanifold through p0 strictly contained in M with the same CR dimension as M. By a theorem of Tumanov, if M is minimal, every continuous CR-function f on M near p extends continuously to a holomorphic function into a wedge W with edge M. Hence we have the following corollary. 0

Corollary 3. Let M  CN , M 0  CN be smooth generic submanifolds of CN and C , respectively, p0 2 M and p00 2 M 0 , M minimal at p0 , H : M ! M 0 a C k0 map which is k0 -nondegenerate at p0 . Then H is smooth in some neighbourhood of p0 . N0

Note that by a regularity theorem of Rosay ([13], see also [1]), if the boundary value of a holomorphic function in a wedge W with edge M is C k on M, then the extension is also of class C k up to the edge. Hence, for the proof of Theorem 2 we will assume that H extends in a C k0 -fashion to a wedge W centered at p0 . We would like to mention one particular instance of this theorem. If M is a manifold whose identity map is k0 -nondegenerate in the sense of Definition 1, then we say that M is k0 -nondegenerate. This notion has been introduced for hypersurfaces by Baouendi, Huang and Rothschild in [2]; for a thorough introduction to this nondegeneracy condition for submanifolds and its connection with holomorphic nondegeneracy in the sense of Stanton ([15]), see [1], or the paper of Ebenfelt [5]. In particular, every CR-diffeomorphism of class C k0 of a k0 -nondegenerate submanifold is k0 -nondegenerate in the sense of Definition 1. Theorem 2 implies the following regularity result for k0 -nondegenerate smooth submanifolds. 0

Corollary 4. Assume that M  CN and M 0  CN are k0 -nondegenerate smooth submanifolds of real codimension d, M minimal at p0 , and H : M ! M 0 is a CR-diffeomorphism of class C k0 . Then H is smooth.

A C1-Regularity Theorem for Nondegenerate CR Mappings

317

If d ¼ 1, we can drop the assumption of minimality, since in the hypersurface case, k0 -nondegeneracy implies minimality. In the case where N ¼ N 0 ¼ 2 and d ¼ 1, Corollary 4 is basically contained in the thesis of Roberts [12]. The case of a Levi-nondegenerate hypersurface is well understood; the connection with the results proved in this paper is that Levi-nondegeneracy of hypersurfaces is equivalent to 1-nondegeneracy. In fact, for Levi-nondegenerate hypersurfaces, Corollary 4 is due to Nirenberg, Webster and Yang [10], and of course we should not forget to mention Fefferman’s mapping theorem [6]. However, let us stress that in this paper, we shall always assume a certain smoothness a priori. A proof for strictly pseudoconvex hypersurfaces of finite smoothness was given by Pinchuk and Khasanov [11]. More recently, Tumanov [16] has proved the corresponding theorem for Levi-nondegenerate targets of higher codimension. For results for pseudoconvex targets, we want to refer the reader to the historical discussion in the paper by Coupet and Sukhov [4] and the newer results for convex hypersurfaces by Coupet, Gaussier and Sukhov [3]. The paper is organized as follows. In Section 2, 3, and 4 we present the technical foundations for the proof. Although these results are well known, they are not easy to find in the literature; so, in order to make this paper as self contained as possible, we have decided to include the proofs. Theorem 2 is then proved in Section 5. 2. Boundary Values of Functions of Slow Growth In this section, we will develop an integral representation for a @-bounded function of slow growth (in a wedge with straight edge). Let us first fix notation. Let U  Cn , V  Rd be open subsets, and let  ¼ ð1 ; . . . ; d Þ 2 Rd with 0 < j for 0 4 j 4 d. We set þ ¼ fðz; s; tÞ 2 U  V  Rd : 0 < t < g,  ¼ fðz; s; tÞ 2 U  V  Rd : 0 > t >  g and 0 ¼ U  V  f0g, and we will write z ¼ ðx; yÞ for the underlying real variables. Throughout the paper, dm will denote Lebesgue measure. Let Bðþ Þ be the space of all functions h 2 C 1 ðþ Þ that extend smoothly to  þ : t 6¼ 0g which have the following property: For each the set E ¼ fðz; s; tÞ 2  compact set K  U  V, there exist positive constants C1 ,  and C2 (depending on K and h) such that sup

jtj jhðx; y; s; tÞj 4 C1

ð2Þ

ðz;sÞ 2 K;0 < t < 

and j@j hðx; y; s; tÞj 4 C2 ;

sup

1 4 j 4 d:

ð3Þ

ðz;sÞ 2 K;0 < t < 

Here we write @j ¼ 12 ð@s@ j þ i @t@ j Þ. We have the following (probably well known) result, which we state for Bðþ Þ; however, we define Bð Þ in a similar manner, and all the results stated in this section hold equally well for Bð Þ. Theorem 5. Let h 2 Bðþ Þ. Then the limit ð hbþ h; i ¼ lim hðx; y; s; Þðx; y; sÞ dm ¼ð1 ;...;d Þ!0 U  V

ð4Þ

318

B. Lamel

exists for each  2 Cc1 ð0 Þ and defines a distribution bþ h called the boundary value of h. Furthermore, for each compact set K there exists an integer v0 such that for v 5 v0, for each j ¼ 1; . . . ; d, 0 4 0 4 j we have the following integral representation for  2 Cc1 ðU  VÞ with supp   K: hbþ h; i ¼

ð

hðx; y; s; 0; . . . ; 0 ; . . . ; 0ÞSv ðx; y; s; 0; . . . ; 0 ; . . . ; 0Þ dm

U V

þ 2i

ð 0

ð U V

@1 hðx; y; s; 0; . . . ; tj ; . . . ; 0Þ

0

 Sv ðx; y; s; 0; . . . ; tj ; . . . ; 0Þ dtj dm ð ð 0 v þ 2i hðx; y; s; 0; . . . ; tj ; . . . ; 0ÞDvþ1 sj ðx; y; sÞtj dtj dm: U V

ð5Þ

0

where Sv ðx; y; s; tÞ ¼

X 1 Ds ðx; y; sÞt : ! jj 4 v

ð6Þ

Proof. Let Sv  be defined by (6). We are going to prove the formula under the assumption that j ¼ 1. Fix ðx; yÞ, s2 ; . . . sd and 0 < 0 < 1 , and assume 0 < 1 < 1  0 . First we are going to assume that K ¼ supp  is contained in a product of the form U1  ½a; b  ½a2 ; b2       ½ad ; bd  contained in a relatively compact open subset W  U  V. In this case, define uðs1 ; t1 Þ ¼ hðx; y; s1 ; s2 ; . . . ; sd ; 1 þ t1 ; 2 ; . . . ; d ÞSv ðz; s; t1 ; 0; . . . ; 0Þ: Clearly, u is C1 on the square ! ¼ ½a; b  ½0; 0  and uðs1 ; t1 Þ ¼ 0 if s1 5 b or s1 4 a. By Stokes formula, ð ð uðs1 ; t1 Þ dw ¼ 2i @uðs1 ; t1 Þ dm; @!

!

where we have set w ¼ s1 þ it1 and @ ¼ @1 . This formula translates into ðb hðx; y; s; Þðx; y; sÞ ds1 a ðb ¼ hðx; y; s; 1 þ 0 ; 2 ; . . . ; d ÞSv ðx; y; s; 0 ; 0; . . . ; 0Þ ds1 a

þ 2i þ 2i

ð 0 ð b 0 ð 0

a ðb

0

a

@1 hðx; y; s; 1 þ t1 ; 2 ; . . . ; d ÞSv ðx; y; s; t1 ; 0; . . . ; 0Þ ds1 dt1 v @1 hðx; y; s; 1 þ t1 ; 2 ; . . . ; d ÞDvþ1 s1 ðx; y; sÞt1 ds1 dt1 :

ð7Þ

A C1-Regularity Theorem for Nondegenerate CR Mappings

319

We integrate this formula with respect to ðx; y; s2 ; . . . ; sd Þ to obtain ð hðx; y; s; Þðx; y; sÞ dm W ð ¼ hðx; y; s; 1 þ 0 ; 2 ; . . . ; d ÞSv ðx; y; s; 0 ; 0; . . . ; 0Þ dm W

þ 2i

ð ð 0 W

þ 2i

ð

W

0 ð 0 0

@1 hðx; y; s; 1 þ t1 ; 2 ; . . . ; d ÞSv ðx; y; s; t1 ; 0; . . . ; 0Þ dt1 dm v hðx; y; s; 1 þ t1 ; 2 ; . . . ; d ÞDvþ1 s1 ðx; y; sÞt1 dt1 dm:

ð8Þ

For each of these integrals, we can use the bounded convergence theorem to take the limit as  ! 0, provided that we choose v 5 K , where K denotes the least integer  for which (2) holds on K and to obtain an estimate of the form jhbþ h; ij 4 Ckkvþ1 (where kkk ¼ maxx 2 U  V;jj 4 k j ðxÞjÞ. Now we pass to the case of general K by covering with finitely many sets of the form considered above and using a partition of unity. The details are easy and left to the reader. & Consider now the class Aðþ Þ of functions h which are smooth on E with the property that for all ,  we have that Dx;y Ds h 2 Bðþ Þ. If h 2 Aðþ Þ, for K  U  V we let l ðh; KÞ the smallest integer  such that sup ðz;sÞ 2 K;0 < t < 

jtj jDx;y Ds hðx; y; s; tÞj 4 C1 ;

jj þ jj 4 l

ð9Þ

for some constant C1 . Let us also introduce the space A1 ðþ Þ of functions in Aðþ Þ with the additional property that for any compact set K  U  V, for any multiindeces  and , and for any nonnegative integer k there exists a constant C such that sup ðz;sÞ 2 K;0 < t < 

jDx;y Ds @j hðx; y; s; tÞj 4 Cjtjk ;

1 4 j 4 d:

ð10Þ

Of course, we define the spaces Að Þ and A1 ð Þ analogously, and the results stated below for Aðþ Þ and A1 ðþ Þ also hold for Að Þ and A1 ð Þ. This can be seen most easily by noting the following useful fact: If hðx; y; s; tÞ 2 Aðþ Þ (or A1 ðþ Þ, respectively), hðx; y; s; tÞ 2 Að Þ (or A1 ð Þ, respectively). We will also need the space of functions which are almost holomorphic on U  V. This is the space AHðU  VÞ ¼ fa 2 C1 ðU  V  Rd Þ : D D D @j aðx; y; s; 0Þ ¼ 0; 1 4 j 4 dg: x;y

s

t

ð11Þ Lemma 6. Let h 2 Aðþ Þ, a 2 AHðU  VÞ, and set a0 ðx; y; sÞ ¼ aðx; y; s; 0Þ. Then ah 2 Aðþ Þ, and bþ ah ¼ a0 bþ h in the sense of distributions. Furthermore, if h 2 A1 ðþ Þ, so is ah.

320

B. Lamel

Proof. By the Leibniz rule, Dx;y Ds ah is a sum of products of derivatives of a and h. It is clear that such a sum fulfills (2). To see that it also fulfills (3), note that by (11) every derivative of @j a vanishes to infinite order on t ¼ 0. To see that bþ ah ¼ a0 bþ h we use Taylor expansion to write X 1 Ds aðx; y; s; 0ÞðitÞ þ Oðjtjkþ1 Þ aðx; y; s; tÞ ¼ ! jj 4 k ðuniformly on compact subsets of U  VÞ: Now choose k 5 0 ðh; KÞ and substitute into (4) for  with supp   K. The claim follows now by taking the limit and using Theorem 5. & Basically the same proof shows the following Lemma. Lemma 7. Assume that X is a vector field on U  V  Rd which is tangent to all subspaces of the form t ¼ c, where c 2 Rd is a constant vector, and such that all the coefficients of X are in AHðU  VÞ. Set X0 ¼ Xjt¼0 . If h 2 Aðþ Þ, then Xh 2 Aðþ Þ, and bþ Xh ¼ X0 bþ h in the sense of distributions. Furthermore, if h 2 Aðþ Þ, so is Xh. 3. An Almost Holomorphic Edge-of-the-Wedge Theorem The main result of this section is the following theorem. Our presentation follows closely [12], but we also want to refer the reader to [14]. We keep the notation from the proceeding section and since we shall use the Fourier transform we also introduce the following new variables: 2 Rn , 2 Rn ,  2 Rd . For a distribution  on U  V we will write ^ð ; ; Þ ¼ h; expðiðx þ y þ sÞÞi for its Fourier transform. Theorem 8. Assume that hþ 2 Aðþ Þ, h 2 Að Þ, and that bþ hþ ¼ b h ¼ h. Then h is smooth. The proof follows from the next Lemma. Lemma 9. Let h 2 Aðþ Þ, and  2 Cc1 ðU  VÞ. Then for every k 2 N there exists a constant Ck such that if ¼ ð ; ; Þ 2 Rn  Rn  Rd with j 4 0 for some j, 1 4 j 4 d, then Ck d jb h 4 : ð12Þ þ hð Þj ð1 þ j j2 Þk Here, Ck depends on k, , and h. The same result holds with Aðþ Þ replaced by Að Þ if j 5 0 for some j. Proof. For the moment, fix ; for simplicity, assume that j ¼ 1, so that 1 4 0. We shall write aðx; y; s; tÞ ¼ expðiðx þ y þ sÞ þ tÞ. Then a 2 AHðU  VÞ – in fact, @j a ¼ 0, 1 4 j 4 d. We let  be the real Laplacian in the 2n þ d variables ðx; y; sÞ, that is, n n d X @2 X @2 X @2 ¼ þ þ : ð13Þ @x2j @y2j @s2j j¼1 j¼1 j¼1

A C1-Regularity Theorem for Nondegenerate CR Mappings

321

We then have that ð1 þ Þk aðx; y; s; tÞ ¼ ð1 þ j j2 Þk aðx; y; s; tÞ. Recall that we d write a0 ðx; y; sÞ ¼ aðx; y; s; 0Þ. By Lemma 6, we see that b h ¼ hbþ h; þ hð Þ a0 i ¼ hbþ h; a0 i ¼ ha0 bþ h; i ¼ hbþ ah; i. We apply the integral formula (5) from Theorem 5 for j ¼ 1, and some 0 , which implies that hbþ ah; i ð ¼

0

hðx; y; s; 0 ; 0Þeiðx þy þsÞ e 1 Sv ðx; y; s; 0 ; 0Þ dm

U V

þ 2i

ð 0

ð U V

þ 2i

ð

U V

ð@1 hðx; y; s; t1 ; 0ÞÞeiðx þy þsÞ et1 1 Sv ðx; y; s; t1 ; 0Þ dt1 dm

0

ð 0 0

v hðx; y; s; t1 ; 0Þeiðx þy þsÞ et1 1 Dvþ1 s1 ðx; y; sÞt1 dt1 dm

¼ I1 þ I2 þ I3 :

ð14Þ

iðx þy þsÞ

k iðx þy þsÞ

1 ð1þj j2 Þk

by ð1 þ Þ e in all three integrals We now replace e above. Then we integrate by parts and estimate, where we choose v 5 2k ðh; KÞ (see (9) for the definition of this number) with K ¼ supp . Since all the estimates are easy, we do not write them out; the reader can easily check them. & Proof of Theorem 8. Let p 2 U  V. Choose a function  2 Cc1 ðU  VÞ which is equal to 1 in some open neighbourhood of p. By Lemma 9, since hþ 2 Aðþ Þ and h 2 Að Þ, we have that c jh hð Þj 4

Ck ð1 þ j j2 Þk

:

ð15Þ

for all 2 R2nþd . Hence, h is smooth (see for Example [7]), and so h is smooth in some neighbourhood of p, since   1 there. Since p was arbitrary, the claim follows. & 4. A Version of the Implicit Function Theorem We will need the following, ‘‘almost holomorphic’’, implicit function theorem. Theorem 10. Let U  CN be open, 0 2 U, A 2 Cp , F : U  Cp ! CN be smooth in the first N variables and polynomial in the last p variables, and assume that Fð0; AÞ ¼ 0 and FZ ð0; AÞ is invertible. Then there exists a neighbourhood U 0  V 0 of ð0; AÞ and a smooth function  : U 0  V 0 ! CN with ð0; AÞ ¼ 0, such that if FðZ; Z ; WÞ ¼ 0 for some ðZ; WÞ 2 U 0  V 0 , then Z ¼ ðZ; Z ; WÞ. Furthermore, for every multiindex , and each j, 1 4 j 4 N, D

@j ðZ; Z ; WÞ ¼ 0; @Zk

1 4 k 4 N;

ð16Þ

if Z ¼ ðZ; Z ; WÞ, and  is holomorphic in W. Here, D denotes the derivative in all the real variables. Proof. Let us write FðZ; Z ; WÞ ¼ Fðx; y; WÞ where ðx; yÞ 2 RN  RN are the underlying real coordinates in CN , as usual identified by Zj ¼ xj þ iyj. Let us also

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B. Lamel

choose a neighbourhood U0  RN of 0 with the property that U0  U0  U. We extend F in the first 2N variables almost holomorphically; that is, we have a ~ : U0  RN  U0  RN  Cp ! CN with the property that function F ~ ðx; x0 ; y; y0 ; WÞjx0 ¼y0 ¼0 ¼ Fðx; y; WÞ F

ð17Þ

and, if we introduce complex coordinates k ¼ xk þ ix0k , k ¼ yk þ iy0k , 1 4 k 4 N, then

~
~
 @ Fj
 @ Fj
D 
¼D ¼ 0; 1 4 j; k 4 N: ð18Þ @ k
x0 ¼y0 ¼0 @ k x0 ¼y0 ¼0 ~ is still polynomial in W. We introduce new coordinates  ¼ Also, F ð1 ; . . . ; N Þ 2 CN by zk þ k zk  k ;

k ¼ ; 1 4 k 4 N; k ¼ 2 2i ; WÞ ¼ Fð ; ; ; ; WÞ. G is smooth in the first 2N complex and write GðZ; Z ; ;  variables in some neighbourhood of the origin, and polynomial in W. We will now compute the real Jacobian of G with respect to Z at ð0; AÞ. At ð0; AÞ, @G @Z ð0; AÞ ¼ @F @G @Z ð0; AÞ and @ Z ð0; AÞ ¼ 0, so that we have 0 1 @G @G

2

B @Z @ Z C Cð0; AÞ ¼
det @F ð0; AÞ
6¼ 0 detB

@ @G  @G A @Z @Z @ Z by assumption. Hence, by the implicit function theorem, there exists a smooth function defined in some neighbourhood of ð0; AÞ, valued in CN , such that Z ¼ ; WÞ solves the equation GðZ; Z ; ;  ; WÞ ¼ 0 uniquely. Here we have ð;  already taken into account that depends holomorphically on W, a fact that the reader will easily check. Since GðZ; Z ; Z ; Z; WÞ ¼ FðZ; Z ; WÞ, this implies that if FðZ; Z ; WÞ ¼ 0, then Z ¼ ðZ ; Z; WÞ. We let ðZ; Z ; WÞ ¼ ðZ ; Z; WÞ and claim that  satisfies (16). In fact, com 1  Z Þ1 ðG þ GZ G  1   Þ, putation shows that Z ðZ; Z ; WÞ ¼  ¼ ðGZ  GZ G G G Z Z where the right hand side is evaluated at ð ðZ ; Z; WÞ; ðZ ; Z; WÞ; Z ; Z; WÞ. This formula shows that each j;Zk is a sum of products each of which contains a factor . which is a derivative of G with respect to Z or  By the definition of G, we have that ~ 1 @F ~ @G 1 @ F ¼ ; þ @ Z 2 @  2i @ 

~ 1 @F ~ @G 1 @ F ¼ :   2 @  2i @  @

By (18) every derivative of those vanishes if x0 ¼ y0 ¼ 0, which is in turn the case   Z if Im ðZ ;ZÞþ ¼ 0 and Im ðZ ;ZÞZ ¼ 0. But this is clearly fulfilled if Z ¼ ðZ ; ZÞ. 2 2i The proof is now finished by applying the Leibniz rule, the chain rule and the observations made above. &

A C1-Regularity Theorem for Nondegenerate CR Mappings

323

Note that it is clear from the usual implicit function theorem that we can solve for N of the real variables ðx; yÞ. What this theorem asserts is that we can do so in a special manner. 5. Proof of Theorem 2 Let us start by choosing coordinates. There is a neighbourhood U of p0 ¼ 0 in CN and a smooth function  : Cn  Rd ! Rd defined in a neighbourhood V of 0 such that M \ U ¼ fðz; s þ iðz; z; sÞÞ : ðz; sÞ 2 Vg with the property that rð0Þ ¼ 0. Since the conclusion of the theorem is local, we shall replace M by M \ U, and use this representation. For suitably chosen open sets U  Cn and V  Rd , consider the diffeomorphism  : U  V ! M, ðz; z; sÞ ¼ ðz; s þ iðz; z; sÞÞ. We extend this diffeomorphism almost holomorphically to a map, again denoted by , from U  V  Rd to CN .  is a diffeomorphism in an open neighbourhood of U  V  f0g, and it has the property that for every component l of , D @j l ðz; s; 0Þ ¼ 0; ðz; sÞ 2 U  V; ð19Þ x;y;s;t

where the derivative is in all the real variables. Equivalently, Dx;y Ds @j l ðz; s; 0Þ ¼ Oðjtj1 Þ; ðz; sÞ 2 U  V;

ð20Þ

uniformly on compact subsets of U  V. That is, for each , , K  U  V compact and every l 2 N there exists a constant Cl ¼ Cl ð; ; KÞ such that jDx;y Ds @j l ðz; s; tÞj 4 Cl jtjl ;

ðz; sÞ 2 K:

ð21Þ

We assume that each component Hj of H extends continuously (and, consequently by a theorem of Rosay [13] already alluded to above, in a C k -fashion) to a holomorphic function into a wedge with edge M. Let us recall that this means that with an open convex cone
in Rd each Hj extends continuously to the set W
¼ fZ 2 U0 :
ðZ; Z Þ 2
g, where U0 is an open neighbourhood of 0 in CN . By choosing
accordingly, and possibly shrinking U0 , we can in addition assume that each Hj is continuous and bounded on the closure of W
, and in fact smooth up to bW
nM. There exists another open, convex cone
0 , relatively closed in
, neighbourhoods U 0  U and V 0  V of 0 2 Cn and 0 2 Rd , respectively, and  ¼ ^
0 ¼ fðz; s; tÞ 2 U 0  V 0 
0 : 0 < t < g ð1 ; . . . ; d Þ > 0 such that the wedge W 0 0 ~ ^
0 Þ  W
. Hence, hj ¼ Hj   is well 0 with flat edge U  V satisfies W
¼ ðW ^ ^
0 and is smooth up to 0 defined on W
for 1 4 j 4 d, extends continuously to W 0 0 ^ 0 bW
nU  V . Since the conclusion of the theorem is local, we can replace U by U 0 and V by V 0. Furthermore, by shrinking the neighbourhoods once more if necessary, we have that there exist positive constants C1 and C2 such that (here, dðA; BÞ denotes the distance between a compact set A and a closed set B) ^
0 Þ 4 dððz; s; tÞ; bW ~
0 Þ 4 C2 dððz; s; tÞ; bW ^
0 Þ: C1 dððz; s; tÞ; bW

ð22Þ

Our next claim is that we can replace
0 by the standard cone Rdþ ¼ ft 2 Rd : t > 0g. In fact, since
0 is open, we can find d linearly independent vectors v1 ; . . . ; vj in
0 . The linear mapping T which maps vj to the j-th standard

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B. Lamel

basis vector ej is invertible, and T 1 ðRdþ Þ 
0 by convexity. Then we can make a complex linear change of coordinates by setting ðz0 ; s0 ; t0 Þ ¼ ðz; T 1 s; T 1 tÞ. Since this coordinate change is linear and there exist positive constants C1 and C2 with C1 jtj 4 jt0 j 4 C2 jtj, (19), (20), and (21) also hold in the new coordinates. We need just one more coordinate change. Claim 1. There exist a number  > 0, coordinates ðz; s; tÞ and positive constants ~
0 for ðz; sÞ 2 U  V, 0 < t <  and C1 jtj 4 C1 and C2 such that ðz; s; tÞ  W ~ 0 dðbW
; ðz; s; tÞÞ 4 C2 jtj for ðz; sÞ 2 U  V, 0 4 t 4 . Proof. Let ej denote the j-th standard basis vector in Rd , 1 4 j 4 d. If d d t ¼ ðt1 ; . . . ; td Þ 2 Rdþ P, then clearly dðt; bRþ Þ ¼ minj¼1 tj. For  > 0 consider the vectors vj ¼ ej þ  l6¼j el , 1 4 j 4 d. For  small enough, these are linearly independent. We now P consider the linear change P of coordinates given by z0 ¼ z, d 0 0 0 0 0 0 0 t ¼ ðt1 ; . . . ; td Þ 7! j¼1 tj vj , s ¼ ðs1 ; . . . ; sd Þ 7! dj¼1 s0j vj . By (22) it is enough to show that there exist positive constants C1 and C2 such that C1 jt0 j 4 dðt; bRdþ Þ 4 C2 jt0 j. The existence of C2 is clear. But if  < 1, then P dðt; bRdþ Þ ¼ mindj¼1 tj ¼ mindj¼1 ðtj0 þ  l6¼j tl0 Þ 5 ðt10 þ . . . td0 Þ 5 d jtj. An appropriate choice for  finishes the argument. & We are going to use the notation introduced in Section 2; that is, we let þ ¼ U  V  ft 2 Rdþ : 0 < t < g. We let hj ¼ Hj   on þ . Claim 2. hj 2 A1 ðþ Þ for 1 4 j 4 N 0. Proof. By all the choices above, hj satisfies the smoothness assumptions. Let us first check that every derivative of hj is of slow growth. Since Hj is holomorphic in ~
0 and continuous on its closure, the Cauchy estimates imply that we have an W estimate of the form ~
0 ÞÞjj j@  Hj ðZÞj 4 C ðdðZ; bW

ð23Þ

jj

@  for each , where @  denotes @Z  . By the chain rule, Dx;y;s hj ðz; s; tÞ is a sum of products of derivatives of  (which are bounded) and a derivative of Hj with respect to Z, evaluated at ðz; s; tÞ, of order at most jj. Hence, by (23) and claim 1 we conclude that there exists a positive constant C such that

jDx;y;s hj ðz; s; tÞj 4 C jtjjj :

ð24Þ

We 0 now have to estimate the derivatives of @m hj for 1 4 m 4 d. But @m hj ¼ P N @Hj   l¼1 @Zl @m l . Hence, if we take an arbitrary derivative of @m hj , we get a sum of products of derivatives of components of  and a derivative of Hj with respect to Z each of which contains a term of the form @m l . By (23) and (21) we conclude that for each compact set K  U  V and each k 2 N there exists a positive constant Ck with jDx;y;s @m hj ðz; s; tÞj 4 Ck jtjk . This proves claim 2. & We now equip U  V with the CR-structure of M; that is, a basis of the CRvector fields near 0 is given by j ¼ 
Lj for 1 4 j 4 n. We almost holomorphically extend the coefficients of the j to get smooth vector fields on an open subset of Cn  Rd  Rd containing 0.

A C1-Regularity Theorem for Nondegenerate CR Mappings

325

Claim 3. For each j, 1 4 j 4 N 0 , there exists a smooth function j ðZ 0 ; Z 0 ; WÞ defined in an open neighbourhood of ð0; ð hð0ÞÞjj 4 k0 Þ in CN  CKðk0 Þ (Kðk0 Þ denoting N 0 jf : jj 4 k0 gj) such that hj ðz; s; 0Þ ¼ j ðhðz; s; 0Þ; hðz; s; 0Þ; ð hðz; s; 0ÞÞjj 4 k0 Þ;

ð25Þ

here, we write h ¼ ðh1 ; . . . ; hN 0 Þ. Furthermore, after possibly shrinking U and V, the right hand side of (25) defines a function in Að Þ. This last claim of course establishes Theorem 2; since hj 2 Aðþ Þ by Claim 2 and by Claim 3 hj 2 Að Þ, we can apply Theorem 8 to see that hj is smooth. Proof. By the chain rule, we have smooth functions l; ðZ 0 ; Z 0 ; WÞ for jj 4 k0 , 1 4 l 4 d 0 , defined in a neighbourhood of f0g  CKðk0 Þ in CN  CKðk0 Þ , polynomial in the last Kðk0 Þ variables, such that 
0 ðh;  hÞðz; s; 0Þ ¼ l; ðhðz; s; 0Þ; hðz; s; 0Þ; ð hðz; s; 0ÞÞ Þ; ð26Þ jj 4 k0

l

Þj ¼ l;;Z 0 ð0; 0; ð hð0; 0; 0ÞÞ and 
0l;Z 0 ðh; h 00 jj 4 k0 Þ. By Definition 1 we can 0 1 choose  ; . . . ; N and l1 ; . . . ; lN such that if we set  ¼ ðl1 ;1 ; . . . ; lN 0 ;N 0 Þ, then Z 0 ð0Þ is invertible. Hence, we can apply Theorem 10; let us call the solution . Then j satisfies (25), and we shrink U and V and choose  in such a way that gj ðz; s; tÞ ¼ j ðhðz; s; tÞ; hðz; s; tÞ; ð hðz; s; tÞÞjj 4 k0 Þ is well defined and   . It is easily checked that gj is a function continuous in a neighbourhood of  in Að Þ as a consequence of (16) and the fact that each hj 2 A1 ðþ Þ. First note that this implies hj ðz; s; tÞ 2 A1 ð Þ, and by Lemma 7,  hj ðz; s; tÞ 2 A1 ð Þ for each . Now, each derivative D of gj is a sum of products of derivatives of j (which are uniformly bounded on  ) and derivatives of h, h, and  h, all of which fulfill the analog of (2) on  . So gj fulfills the analog of (9) on  . Next,  k . We have that we compute the derivative of gj with respect to w N0 N0 X X @ @ h @gj @j @hl X @j @ hl ¼ þ þ : k k  k l¼1 @ Zl0 @ w  k jj 4 k @W @ w @w @Zl0 @ w l¼1 0

Applying any derivative D , we see that the first sum gives rise to products of @ derivatives of @Zj0 and derivatives of h, h, and  h. Now the derivatives of j fulfill l (16). Since on t ¼ 0, h ¼ ðh; h; ð hÞjj 4 k0 Þ, we conclude that h  ðh; h; @ ð  hÞjj 4 k0 Þ ¼ OðjtjÞ. But by (16), any derivative of @Zjl ðZ; Z ; WÞ is OðjZ @ 1 j ðZ; Z ; WÞj Þ, so that derivatives of @Z 0 evaluated at ðh; h; ð hÞjj 4 k0 Þ are l Oðjtj1 Þ. All the other terms in the product are Oðjtjs Þ for some s, so that the terms coming from the first sum are actually Oðjtj1 Þ. For the second and third sum, a similar argument using that h and  h are in A1 ð Þ implies that all the terms arising from them are Oðjtj1 Þ. All in all, we conclude that gj 2 A1 ð Þ, which finishes the proof. & References [1] Baouendi MS, Ebenfelt P, Rothschild LP (1999) Real submanifolds in complex space and their mappings. Princeton, NJ: Univ Press [2] Baouendi MS, Huang X, Rothschild LP (1996) Regularity of CR mappings between algebraic hypersurfaces. Invent Math 125: 13–36

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